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Proving that Z2 X Z2 X Z2... Z2 is a isomorphic (ring isomorphism) to P(N)
I wish to prove that the ring of Cartesian product Z2 X Z2 X Z2...X Z2 (here we have n products) under addition and multiplication (Z2 is {0,1}) is isomorphic to P(N) where P(N) is the ring of power set of the set of n elements where the multiplication operation is AB=A U B and the addition operation is A+B=(A U B)-(A[itex]\cap[/itex]B)
Both of these rings are Boolean rings
A ring isomorphism is an operation preserving (both operations) one to one and onto map between the two rings
I tried for smaller cases such as Z2 => P(1) and I was able to find an isomorphism through brute force. I want to somehow use the fact that these two rings are Boolean rings to solve this.
While searching for a solution I ran into something called "the Stone Representation theorem" which states that all Boolean Algebras are isomorphic to a field of sets. Is it possible to apply this theorem in this case?
Thank you for the help
Homework Statement
I wish to prove that the ring of Cartesian product Z2 X Z2 X Z2...X Z2 (here we have n products) under addition and multiplication (Z2 is {0,1}) is isomorphic to P(N) where P(N) is the ring of power set of the set of n elements where the multiplication operation is AB=A U B and the addition operation is A+B=(A U B)-(A[itex]\cap[/itex]B)
Homework Equations
Both of these rings are Boolean rings
A ring isomorphism is an operation preserving (both operations) one to one and onto map between the two rings
The Attempt at a Solution
I tried for smaller cases such as Z2 => P(1) and I was able to find an isomorphism through brute force. I want to somehow use the fact that these two rings are Boolean rings to solve this.
While searching for a solution I ran into something called "the Stone Representation theorem" which states that all Boolean Algebras are isomorphic to a field of sets. Is it possible to apply this theorem in this case?
Thank you for the help